Question: All of the 3rd grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$2.00$ each for students, and the group paid $$33.00$ in total. The next month, the same group visited a science museum where the tickets cost $$22.50$ each for teachers and $$12.50$ each for students, and the group paid $$157.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+2y = 33}$ ${22.5x+12.5y = 157.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-6y = -99}$ ${22.5x+12.5y = 157.5}$ Add the top and bottom equations together. $ 6.5y = 58.5 $ $ y = \dfrac{58.5}{6.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7.5x+2y = 33}$ to find $x$ ${7.5x + 2}{(9)}{= 33}$ $7.5x+18 = 33$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {22.5x+12.5y = 157.5}$ and get the same answer for $x$ ${22.5x + 12.5}{(9)}{= 157.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.